0
votes
1answer
42 views

Use complex number to solve this equation $\int e ^{3x} cos x dx$?

I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
1
vote
1answer
19 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
4
votes
4answers
48 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $\int_{-\infty}^\infty dk_0 ...
1
vote
0answers
15 views

Calculate $\frac{1}{2\pi}\int_{-\pi}^{\pi} {\rm{sinc}}(\alpha-t)\ e^{-\imath z t} dt$

Let us consider the $ {\rm{sinc}}$ normalized function: \begin{equation} {\rm{sinc}}(\tau)= \begin{cases} \frac{ \sin(\pi \tau)}{\pi \tau} \qquad &\tau \not= 0,\\ 1\qquad & \tau=0, ...
0
votes
2answers
90 views

Smart way to Integrate $\int \dfrac{dx}{1+x^4}$ [duplicate]

I know abt that straightforward lengthy way. anyone knows a smarter way
1
vote
0answers
26 views

Integrate complex function

In a physics textbook I am working with, the following integral is caluclated: Define the complex elastic modulus as $\overline{M} = M_1 + i M_2$ where $\overline{M}$ is a function of the angular ...
1
vote
1answer
47 views

show an integral is bounded by a constant independent of a parameter

This is a question in Treves. Suppose $a>1$ and $\tau \in \mathbb R $, (i) show that for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge(\tau ^2+|\xi|^2+a^2)^{1/2}$ (ii) ...
4
votes
0answers
74 views

Help computing integral of quarter contour

For a project I'm working on I need to solve the following integral. Can anyone help me tackle this challenge? The Integral: $$ \beta \int_{\pi}^{\pi/2} {i k^2 r e^{i \theta} + ikr^2 e^{2i\theta} ...
0
votes
1answer
19 views

A problem on orthonormality of a set of complex functions

The following is a problem of an undergraduate exam test:
1
vote
0answers
63 views

Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
1
vote
1answer
47 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
0
votes
2answers
57 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
0
votes
1answer
85 views

This integral is strange

$$ \int_{C_1}\frac{dz}{z}=\int_0^{2\pi}\frac{-R\sin{t}+iR\cos{t}}{R\cos{t}+iR\sin{t}}dt=\int_0^{2\pi}i\text{ }dt=2\pi i\tag{24.36} $$ Shouldn't it simply be $$\left[\ln(R \cos t + iR \sin ...
3
votes
1answer
69 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
2
votes
0answers
40 views

Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
4
votes
2answers
84 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
0
votes
3answers
141 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
0
votes
0answers
25 views

Complex number contour integral

Determine the contour integral ∫Ѱ 1/z dx, where Ѱ is the positively oriented unit circle with centre at -2 given that Ѱ(t) = -2 + e^(it), 0<=t<=2pii. I understand that Ѱ is the positively ...
0
votes
2answers
86 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
3
votes
2answers
192 views

What insight is supposed to be gained from this complex analysis exercise?

Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this: $$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in ...
0
votes
1answer
42 views

Calculating a complex integral by rewriting as a contour integral on |z|=1.

I need to show that $\int_0^{2\pi}\frac{d\theta}{2+i\:sin\theta}=\frac{2\pi}{\sqrt{5}}$ I used $sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$ and substituted $z=e^{i\theta}$. I ended up with ...
1
vote
1answer
441 views

Complex integration around a branch point

I am confused about the "deformation" of a closed contour that my book is doing. For reference, it is example 2.4.3 on pg. 75-76 from this free online book. The example is the integration of 1/z ...
2
votes
2answers
102 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
3
votes
0answers
58 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
0
votes
1answer
41 views

Integration of a complex number

I'm trying to integrate, for example, $\int e^xe^{-inx} \,dx$. $i$ is the imaginary unit, $n$ is a constant. I tried to integrate normally - as I would in a Real number: $$\int e^x e^{-inx}\,dx = ...
0
votes
2answers
110 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
6
votes
2answers
166 views

Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
0
votes
4answers
81 views

Integration $1/x$ - complex number

Why there is no integral $$\int_{-e}^{e}\frac{1}{x}$$ And why integral $$\int_{-e}^{-1}\frac{1}{x}= -1$$ and not $$\int_{-e}^{-1}\frac{1}{x}=(-1 + i\cdot\pi)$$ E.g. ...
1
vote
1answer
139 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
1
vote
0answers
33 views

Integral over a complex plane

I am wondering if closed-form solutions exist for the following integral: $$\int_\mathbb{C}e^{-\frac{|u-v|^2}{c}-|v|^2}|v|^{2n}d^2v$$ where $u$ is a complex number, $c>1$ a real constant and $n$ ...
3
votes
2answers
77 views

Complex integral of an exponent divided by a linear ($\int \frac{e^u}{u-1}$)

Here is the question I'm working on: Evaluate the following integral: $$ \oint_{|z+1|=1} \frac{\sin \frac{\pi z}{4}}{z^2-1}dz$$ I've tried along the following line. Substitute $sin(z) = ...
1
vote
1answer
84 views

Given $Re(f(z))$ and the fact that $f(z)$ is analytic, find $Im(f(z))$

The question I'm trying to answer: Find an analytic function $f(z)$ whose real part $u(x,y)$ is: $$\frac{y}{x^2+y^2}$$ An analytic function satisfies the Cauchy-Riemann relations. So I thought ...
2
votes
1answer
171 views

when point lies outside what does cauchy integral formula state ??

I have the following complex line integral: $$ \int_{|z| = 2} \frac{z}{z - 3} $$ My prof said it is 0,but did not explain.He just said that the point 3+0*i lies outside the circle. But the cuachy ...
0
votes
2answers
115 views

Integral of complex periodic signal

Integrating $e^{j(k-n)\omega t} \, dt$ over the interval $0$ to $T$ where $T$ is the fundamental time period of the sinusoids yields zero when $k$ is not equal to $n$.... ? how? assume ω is the ...
2
votes
0answers
96 views

Evaluating integral with branch of log

I'm having trouble understanding what this question even means really. "Let $\gamma$ be the semi-circle from $2i$ to $-2i$ that passes through $2$ in the positive direction. Find $\int_\gamma ...
0
votes
0answers
68 views

Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n} $?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
4
votes
1answer
57 views

Integration With i

Why does this approach to integration not work? If there is an integral $1/\sqrt{a^2-x^2}$, the answer is $\arcsin(x/a)$. But if the integral is $1/\sqrt{x^2-a^2}$ then it is $\log(x+\sqrt{x^2-a^2})$. ...
0
votes
1answer
55 views

Contour integration of complex number confuses me, still.

Given $f(z) = (x^2+y)+i(xy)$ and we integrate it using the Parabola Contour. For a parabola, $\gamma(t) = t + it^2$. So, $f(\gamma(t)) = 2t^2 + it^3$. What was ...
0
votes
1answer
61 views

Find the integral of $\overline{z}$

Question: Find $\int\overline{z}$, when the contour is a parabola. Interval is from 0 to 1. My Attempt: $z = x + iy \Rightarrow \overline{z} = x - iy$ $f(z) = x - iy$ Since the contour is a ...
0
votes
2answers
142 views

Difficulty in understanding integrals of complex numbers

I understand what integration of real numbers is. I know how the definition of it is made. I have trouble in understanding how it works for complex numbers. I am referring to the notes here: ...
1
vote
0answers
18 views

simplification of a complex expression

I am collecting proofs for certain integrals. To simplify certain proofs, I use $e^{Ax}cos(Bx)=\mathcal{Re}[e^{(A+iB)x}]$, where $A$, $B$, and $x$ are real. Is there an analogous simplification for $ ...
1
vote
2answers
148 views

Determine the integral $\int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$ using residues.

Determine the integral $$ \int_0^\infty \frac{\mathrm{d}x}{(x^2+1)^2}$$ using residues. This is from Section 79, Brown and Churchill's Complex Variables and Applications. In order to do this. We ...
1
vote
1answer
118 views

$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform

I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple: $$ f(x) = \sin(x) $$ By definition its Fourier transform should be: $$ F(k) = ...
0
votes
1answer
1k views

Using Cauchy's integral formula to evaluate a function

This problem is from Brown/Churchill Complex Variables and Applications, 8th edition 2009. Section 52, exercise 2, subsection (a) How do I show that the integral of the function $g(z) = ...
3
votes
1answer
130 views

Integrating $\sin^2(x)$ using imaginary numbers.

I know I can change "$\sin^2\theta$" to "$\frac{1}{2}(1-\cos(2\theta))$" or use integration by parts, but I was curious about doing it using imaginary numbers. I tried this but it didn't work. $$\int ...
0
votes
2answers
45 views

Short integration involving $e$ and $i$

Durring our lectures our professor calculated an integral like this: $$ \Psi = \Psi_0 \int\limits^{k+\delta k}_{k-\delta k} \! e^{ikx}\, \textrm{d}k = \Psi_0 \int\limits^{k+\delta k}_{k-\delta k} \! ...
4
votes
3answers
785 views

Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$

I'm having trouble with this problem: Consider the integral: $$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$ a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
1
vote
0answers
87 views

Hadamard regularization isn't working out

As part of an exercise in a grad course on "mathematical methods" (always such a helpful name), I've been asked to evaluate $I=\int_0^{1/2}{(x^2-x+c)^{-2}dx}$ as a Hadamard finite part integral for $0 ...
2
votes
1answer
135 views

Algebraic integral involving complex numbers

I need some help analytically proving the following with elementary tools: $$\int_1^{+\infty} \frac{z^i + z^{-i}}{z^2 + 1} ~ \mathrm{d} z = \frac{\pi}{2} \mathrm{sech} \left ( \frac{\pi}{2} \right ...
1
vote
1answer
42 views

Unwanted $i$ floating around when trying to calculate $\langle p\rangle$

$\def\sp#1{\left\langle#1\right\rangle}$I am given $$ \Psi(x,0)=A_0 \exp\left(-\frac{x^2}{2\sigma_0^2}\right) \cdot \exp\left(\frac{i}{\hbar}p_0x\right)\tag1$$ where $A_0=(\pi ...